How do I determine c1/c2 based on the values below?

Question

C1= 4/3 ( cos 5pi/4+ i  sin 5pi/4)

C2=3 (cos pi/4+ i sin pi/4)

Answer 1

To divide complex numbers given in polar form [r(cosθ + isinθ)], divide the r values and subtract the angles.

 

So, C₁ / C₂ = (4/3 ÷ 3)[cos(5π/4 - π/4) + isin(5π/4 - π/4)]

 

                 = (4/9)[cosπ + isinπ] = (4/9)[-1 + 0i] = -4/9

Answer 2

So as to simplify the notation,

LET

 T = 5pi/4= 5 (pi/4)  and x = pi/4 so T = 5x and x-T = pi/4 - 5pi/4 = -pi <---- see below

 

The complex fraction 4/3/3 = 4/3 x 1/3 = 4/9

 

The quotient C1/C2 then becomes 

 

4/( cos T + i sin T)/ [9(cos x + i sin x)]

 

Next, we must rationalize the denominator by multiplying 

the top and bottom by (cos x - i sin x)

 

4(cos x - i sinx)(cos T + i sin T) / 9 ( cos x + i sin x)(cos x - i sin x) ]

 

By FOIL Method and trig identity,

(cos x)^2 - (-1) (sin x)^2 = <--- cross terms cancel

(cos x)^2 + (sin x)^2 = 1

So the denominator is just 9

 

 

The quotient can be written as:

(4/9)(cos x - i sinx)(cos T + i sin T)

 

FOILing again....

(4/9)[ cos x cos T  - (-1) (sin x)(sin T)] <--- cross terms cancel

 

(4/9) cos x cos T + sin x sin T

 

(4/9) cos(x-T)

 

(4/9) cos (-pi) <---- see above]

 

(4/9) cos(pi) <---- cosine is even function, so it eats the negative sign;

                             -pi = pi  = 180 degrees

(4/9)(-1)

 

= -4/9